CS 583: Approximation Algorithms: Covering Problems∗

Packing and Covering problems together capture many important problems in combinatorial optimization. We will consider covering problems in these notes. Two canonical covering problems are Vertex Cover and its generalization Set Cover. They play an important role in the study of approximation algorithms. A vertex cover of a graph G = (V,E) is a set S ⊆ V such that for each edge e ∈ E, at least one end point of e is in S. Note that we are picking vertices to cover the edges. In the Vertex Cover problem, our goal is to find a smallest vertex cover of G. In the weighted version of the problem, a weight function w : V → R+ is given, and our goal is to find a minimum weight vertex cover of G. The unweighted version of the problem is also known as Cardinality Vertex Cover. In the Set Cover problem the input is a set U of n elements, and a collection S = {S1, S2, . . . , Sm} of m subsets of U such that ⋃ i Si = U . Our goal in the Set Cover problem is to select as few subsets as possible from S such that their union covers U . In the weighted version each set Si has a non-negative weight wi the goal is to find a set cover of minimim weight. Closely related to the Set Cover problem is the Maximum Coverage problem. In this problem the input is again U and S but we are also given an integer k ≤ m. The goal is to select k subsets from S such that their union has the maximum cardinality. Note that Set Cover is a minimization problem while Maximum Coverage is a maximization problem. Set Cover is essentially equivalent to the Hitting Set problem. In Hitting Set the input is U and S but the goal is to pick the smallest number of elements of U that cover the given sets in S. In other words we are seeking a set cover in the dual set system. Set Cover is an important problem because it and its special cases not only arise as an explicit problems but also because many implicit covering problems can be cast as special cases. Consider the well known MST problem in graphs. One way to phrase MST is the following: given an edgeweighted graph G = (V,E) find a minimum cost subset of the edges that cover all the cuts of G. This may appear to be a strange way of looking at the MST problem but this view is useful as we will see later. Covering problems have the feature that a superset of a feasible solution is also feasible. More abstractly one can cast covering problems as the following. We are given a finite ground set V (vertices in a graph or sets in a set system) and a family of feasible solutions I ⊆ 2V where I is upward closed; by this we mean that if A ∈ I and A ⊂ B then B ∈ I. The goal is to find the smallest cardinality set A in I. In the weighted case V has weights and the goal is to find a minimum weight set in I. ∗Based on previous scribed lecture notes of Abul Hassan Samee and Lewis Tseng.